CURRICULUM for Computer Science and Mathematics

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1 CURRICULUM for Computer Science and Mathematics MSc SEPTEMBER 2015 ROSKILDE UNIVERSITY 1

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3 This curriculum is issued pursuant to Executive Order no of 16 December 2013 on Bachelor and Master s programmes at the universities (the Universities Order), as subsequently amended, and Roskilde University s Common Rules of 27 September 2012, as subsequently amended. 1. The study programme 1.1 Object 1.2 Competency profile 1.3 Main area affiliation Title 1.6 Study Board 1.7 Affiliation of corps of external examiners 2. Admission requirements and credit 2.1 Admission requirements 2.2 Credit 2.3 Advance credit 3. Programme elements 3.1 Courses 3.2 Projects 3.3 Internship 3.4 Thesis 4. Structure of the programme 4.1 Schematic breakdown 4.2 Recommended course of study 5. The study programme 5.1 First semester 5.2 Second semester 5.3 Third semester 5.4 Fourth semester thesis 6. General regulations 6.1 Special examination facilities 6.2 Academic selection criteria 6.3 Time limit for study programme completion 6.4 Registration for a third attempt in a required subject 6.5 Registration/de-registration and re-examinations 7. Exemptions and appeals 7.1 Exemptions 7.2 Appeals 8. Entry into force and transitional provisions 8.1 Entry into force 8.2 Transitional rules Appendix 1. Automatic administrative registration for study activities 2

4 1. About the study programme The MSc programme in Computer Science and Mathematics is an interdisciplinary study programme consisting of two independent subjects, of which Computer Science comprises subject 1 and Mathematics comprises subject Object The object of the combination Master s programme MSc in Computer Science and Mathematics is to further develop the student s knowledge, skills and competencies within software development and mathematics. The programme s focus is on equipping students with the skills to organise, manage and plan a system development project, including the software part, in which the development and application of mathematical models and methods is particularly central, and where the focus is on formal algorithmic logic, formal languages and computability, with particular emphasis on qualifying graduates for work in the IT industry as IT developers, programmers or IT architects, business developers or project managers, especially in connection with technically specialised software in logistics, operations research or database optimisation. The combination MSc programme also provides graduates with the academic qualifications to commence a PhD programme. The programme is presented in English. Unless otherwise specified, the examination language is the same as the language of instruction. 1.2 Competency profile The combination Master s programme MSc in Computer Science and Mathematics provides the student with the following competencies: : Research-based knowledge of techniques and theories used to analyse, design and construct IT systems, including the methods and principles of software engineering; computer and system architecture; algorithms and data representation; design principles for human-machine interfaces. of what constitutes Computer Science and Mathematics as subjects with respect to the knowledge areas, forms of understanding, focus points and development of a conceptual world, cognition, reasoning and modes of representation and approach that are characteristic of the subjects. Research-based knowledge of theories and methods used in the analysis, design and implementation of IT applications, including the methods and principles of system development; IT architecture; innovative IT applications as product, service or process; interaction between IT solutions and use. Advanced knowledge of selected specialised fields of computer science, on the basis of the highest standard of international research. of geometry, probability theory and statistics, and of the fundamental structures of mathematics, as well as exemplary selected mathematical topics of an advanced nature. : in critically relating to, selecting and utilising methods and techniques for the analysis, design and construction of software systems. in programming advanced software solutions with the use of appropriate, state-of-the-art programming languages, libraries, development tools and equipment. The ability to test, validate and evaluate software systems. The ability to contribute key computer science and mathematical knowledge to collaborative projects of an interdisciplinary nature. in communicating research-based knowledge about computer science and mathematics, and discussing professional IT issues on a scientific basis with both colleagues and non-specialists. : The competency to identify, formalise, analyse, evaluate and solve scientific problems on the basis of Computer Science and Mathematics, independently or in co-operation with others. 3

5 The ability to organise, plan and manage a software development project that is complex and requires new solution models. The competency to independently initiate and implement computer science development work with interdisciplinary co-operation, and to assume professional responsibility. The competency to understand, refine, develop, implement and evaluate experimental studies, simulations, reasoning or evidence. The competency to understand, construct, apply and evaluate qualitative and/or quantitative models in the subjects. The ability to relate in a reflective manner to the ways in which the subjects are applied, and to place this in a wider perspective. The ability to systematically and critically familiarise oneself with new areas of the disciplines, and thereby take independent responsibility for one s own professional development and specialisation. 1.3 Main area affiliation The study programme belongs under the main area of Natural Science. The study programme elements of Mathematics belong under the main area of Natural Science. 1.4 The combination MSc programme in Computer Science and Mathematics is a two-year programme corresponding to Title cand.scient. i Datalogi og Matematik Master of Science (MSc) in Computer Science and Mathematics 1.6 Study Board The programme belongs under the Study Board for Communication, Business & Information Technologies. 1.7 Affiliation of corps of external examiners The programme is affiliated with the corps of external examiners for Computer Science. The study programme elements of Mathematics belong under the corps of external examiners for Mathematics. 2. Admission requirements and credit 2.1 Admission requirements See appendix. 2.2 Credit Roskilde University may in individual cases or in accordance with general rules laid down by the University approve that completed study programme elements, cf. the Education Order, may substitute for elements of another programme at the same level, in accordance with the Education Order. Roskilde University may also approve that completed study programme elements, taken at another Danish or foreign institution of higher education on the same level, may substitute for study programme elements, in accordance with the Education Order. Applicants are required to apply for credit for any previously passed study programme elements at Master s level, cf. the Master s Programme Admission Order. 4

6 2.3 Advance credit Students who, as part of a study programme, wish to take elements of the programme at another university or institution of higher education in Denmark or abroad may apply to Roskilde University for advance credit for the planned programme elements. Approval for advance credit can only be granted if the student, in connection with the application, undertakes to provide the necessary documentation, once the advance-credited programme elements have been taken, of whether or not the elements have been passed. The student must also consent to Roskilde University obtaining the necessary information from the host institution if the student cannot obtain the documentation. Once documentation is available showing that the advance-credited programme elements have been passed, Roskilde University will administratively approve the transfer of credit for these to the study programme at the University. In cases where the advance-credited programme elements are for example not offered by the host institution, Roskilde University may at the request of the student undertake alterations in the approval of advance credit. 3. Programme elements 3.1 Courses These study elements provide the student with theoretical and methodological knowledge as well as skills in the academic field of the programme. The student must be given academic preparation to focus on issues relevant to project work and the thesis. Courses may take the form of lectures, classes, case work, workshops, etc. 3.2 Projects Project work is exemplary, problem-oriented and participant-managed, and is intended to develop and document the student s skills in applying scientific theories and methods to work within a defined academic area. Project work requires the independent formulation of a freely-chosen issue within the framework of the curriculum. The project work takes place in groups of 2-6 students. The topic of the project work is selected by the students from among the topics and themes determined by the Study Board. In special cases, the Head of Studies may grant permission for a student s project to be written or examined individually. 3.3 Internship A student may apply to the Study Board for pre-approval to take a specially-designed, academically relevant, project-oriented internship which substitutes for individual, specified elements of the ordinary study programme. The project-oriented internship, which includes, inter alia, the compilation of an internship project report, corresponds to 15. Once the application for pre-approval has been granted, the Head of Studies will assign a supervisor to the student. In its processing of the application, the Study Board must ensure that the student, through the project-oriented internship, achieves academic competencies which in scope and level correspond to the competencies that can be achieved through the ordinary study programme. 3.4 Thesis The combination Master s programme includes a major independent study which is documented through the compilation of a thesis. In the thesis, the student must document knowledge and skills in applying scientific theories and methods to work on a defined, academic and relevant issue. The student must demonstrate skills in analysing, categorising, discussing, reasoning, assessing and reflecting on a scientific basis, and must be able to select and critically evaluate the sources, literature, theory and methods used in the thesis. Through the thesis, the student must demonstrate the ability to communicate an academic study to peers and demonstrate the competency to initiate, manage and complete a long-term process of academic study and writing. 5

7 4. Structure of the programme 4.1 Schematic breakdown 4.2 Recommended course of study To help to ensure the quality and progression of the study programme and support students in their study planning, the following course of study is recommended: Autumn start First semester (Computer Science): Computer Science Course in Mandatory Core Topics(10 ECTS) Optional course in Computer Science (5 ECTS) Project portfolio in Computer Science (15 ECTS) Second semester (Mathematics): Course in Fundamental Mathematical Structures (10 ECTS) Optional course in Probability Theory and Statistics (5 ECTS) * or Advanced Mathematics (5 ECTS) Reflection project in Mathematics (15 ECTS) Third semester (Computer Science and Mathematics): Computer Science: Three optional courses in Computer Science (3*5 ECTS) Mathematics: Course in Geometry (10 ECTS) * Course in Advanced Mathematics (5 ECTS) Fourth semester: Thesis (30 ECTS) Spring start: First semester (Computer Science): Computer Science Course in Mandatory Core Topics(10 ECTS) Optional course in Computer Science (5 ECTS) Project portfolio in Computer Science (15 ECTS) 6

8 Second semester (Mathematics): Course in Geometry (10 ECTS) * Course in Advanced Mathematics (5 ECTS) Reflection project in Mathematics (15 ECTS) Third semester (Computer Science and Mathematics): Computer Science: Three optional courses in Computer Science (3*5 ECTS) Mathematics: Course in Fundamental Mathematical Structures (10 ECTS) Optional course in Probability Theory and Statistics (5 ECTS) * or Advanced Mathematics (5 ECTS) Fourth semester: Thesis (30 ECTS) * In the case of content overlap between the qualifying Bachelor programme and the Mathematics portion of the Master s programme, students will be referred by the Head of Studies for Mathematics to either Theme Course A, B or C, or a modelling project. 5. The study programme 5.1 First semester Object The object of the first semester is to ensure the student acquires in-depth knowledge of a number of core areas within computer science, including machine architecture and operating systems, distributed systems, databases and humanmachine interaction. Study programme elements in the first semester. The semester encompasses: Course in Mandatory Core Topics(10 ECTS) Optional course in Computer Science (5 ECTS) Project portfolio in Computer Science (15 ECTS) Title Language of instruction Mandatory Core Topics (Computer Science) English Compulsory course 10 ECTS of key theories in the core areas of the subject. of techniques in the core areas of the subject for the design and construction of software systems that satisfy specified requirements. A clear overall impression and understanding of the general principles behind the software and hardware systems used by modern computers, and the user's interaction with these. 7

9 in selecting and applying appropriate methods and techniques from the discipline to analyse, design and construct software systems. Learning outcome/ criteria Overall content Competency to work with computer science issues, both independently and in teams. The ability to systematically and critically familiarise oneself with new approaches in the discipline, and thereby take responsibility for one s own professional development. The core areas within Computer Science consist of the topics of machine architecture and operating systems, distributed systems, databases and human-machine interaction. The course encompasses a presentation, critical discussion and testing of knowledge within key areas of each of these topics. Teaching and working methods Form of examination The specific content is described at kursus.ruc.dk. Lectures with exercises, or classes, etc. Oral examination The oral examination has a duration of 30 minutes, including assessment. Group examination or individual examination Moderation Individual External Title Language of instruction Optional courses in computer science (Computer Science) English Optional courses 5 ECTS of a specific area within the field of computer science. of the techniques of the area for the design and construction of software systems that satisfy specified requirements. A clear overall impression and understanding of the general principles behind the theory, methods and technical solutions of the area. criteria in selecting and applying appropriate methods and techniques from the area to analyse, design and construct reliable and user-friendly software systems. Competency to work with computer science issues, both independently and in teams. Competency to critically and systematically familiarise oneself with new approaches in the discipline, and thereby take responsibility for one s own professional development. 8

10 Overall content Teaching and working methods Optional courses give students an opportunity to specialise in a specific subject area in which they acquire the knowledge, skills and competencies to apply theories, methods and solution ideas to their own practice in software development. Optional courses may deal with: robotics, artificial intelligence, web technologies, programming, parallel computation, mobile computers, etc. Usually class teaching, i.e. a combination of teacher presentations, student presentations, and practical work on specific tasks. Lectures with exercises. See the course descriptions at kursus.ruc.dk. Form of examination Group examination or individual examination Moderation Oral examination The oral examination has a duration of 20 minutes, including assessment. Individual External Title Language of instruction Project portfolio (Computer Science) English Project portfolio 15 ECTS In-depth knowledge of the theoretical, methodological and practical possibilities and problems associated with software development, using specific models from the core areas or optional topics. criteria To describe and reflect upon an independently performed task that addresses a complex issue using relevant solution models. To define and justify a selected solution model and independently plan and implement solutions using relevant scientific literature at a high academic level. To master concepts, theories and methods on the basis of the literature, and to be able to use these in a reflective manner to solve specific problems in computer science. Proficiency in computer science development situations that are complex and require new solution models. Students develop their own project portfolios in groups of 2-6 students. The topics of these must lie within the core areas. Overall content Teaching and working methods Core areas within computer science consist of the topics of machine architecture and operating systems, distributed systems, databases and human-machine interaction. A project portfolio is a collection of works (texts, program code, data models, architecture) at various layers of abstraction, associated with a practice-oriented workshop or exercises. The project portfolio must also include a reflection document which cuts across the core areas. The reflection document must be five standard pages in length at 2,400 characters (incl. spaces) per student in the group, and must link practical experience to relevant theory from the core area of computer science. The portfolio must also be 9

11 Form of examination equipped with a front page, table of contents, bibliography and possible appendices. The project portfolio is normally carried out as a group project with 2-6 participants, but in special cases, the Head of Studies may grant permission for a student to compile the portfolio alone. The topic of the project portfolio work is chosen by the students in accordance with the requirements for learning outcomes and any defined topics and themes laid down by the Study Board. Oral examination based on the project portfolio. Up to five minutes are allocated per student during the examination for a self-chosen, prepared presentation based on the joint project portfolio work. The examination otherwise takes the form of a conversation between the student(s), the examiner and the moderator, under the direction of the examiner. During the examination, it must be ensured that all examinees are given an equal opportunity to demonstrate their knowledge of the joint work and their academic standard in relation to the expected learning outcomes. The duration of the examination, including assessment, is as follows: One examinee: 30 minutes Two examinees: 60 minutes Three examinees: 75 minutes Four examinees: 90 minutes Five examinees: 105 minutes Six examinees: 120 minutes Group examination or individual examination Moderation Group examination Internal 5.2 Second semester Object The object of the semester is to train the students to think like mathematicians and equip them with in-depth insight on a scientific level into one or more fields relating to mathematics, in order to increase the students knowledge, skills and competence at Master s level within mathematics. Study programme elements in the second semester The semester encompasses: Autumn start: Course in Fundamental Mathematical Structures (10 ECTS) Optional course in Probability and Statistics (5 ECTS) * or Advanced Mathematics (5 ECTS) Reflection project in Mathematics (15 ECTS) Spring start: Course in Geometry (10 ECTS) * Course in Advanced Mathematics (5 ECTS) Reflection project in Mathematics (15 ECTS) * In the case of content overlap between the qualifying Bachelor programme and the Mathematics portion of the Master s programme, students will be referred by the Head of Studies for Mathematics to either Theme Course A, B or C, or a 10

12 modelling project. Title Fundamental Mathematical Structures (Mathematics) Compulsory course 10 ECTS Specific mathematical structures within set theory, topology and analysis, algebra and geometry. Common features and differences between such structures. Different types of reasoning and proofs, and their importance. Construction and formalisation of such structures. criteria To recognise fundamental mathematical structures. To know and use symbols and other representations in accordance with the given formalism. To read, understand and reproduce proofs in the context of the structures studied. Overall content Teaching and working methods The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to be able to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to be able to decode and apply mathematical symbolic language within a given formalism, and to assess the strengths and weaknesses of an axiomatic system. The competency to be able to read and understand mathematical texts concerning the basis of the subject and fundamental structures, and to communicate these both orally and in writing. Various fundamental, abstract mathematical structures and their interrelations. Introduction to formal logic, including the concept of a formal theory. Set theory, algebraic structures, metric and topological spaces, geometric structures and aspects of measure spaces. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed via an oral examination. Form of examination Group examination or individual examination Moderation The examinee is given a take-home question three days before the oral examination. At the oral examination, the student first of all provides his or her answer to the oral question in a pre-prepared presentation with a maximum duration of 10 minutes. The student is then questioned about the curriculum generally for around 15 minutes. The total duration of the examination is 30 minutes, including assessment. Individual External 11

13 Title Probability Theory and Statistics (Mathematics) Optional course 5 ECTS General features of discrete and continuous probability distributions The most common discrete and continuous probability distributions. Stochastic variables, independence, mean, variance, co-variance and correlation of stochastic variables. Statistical models, maximum likelihood principle, estimator tests, resampling and non-parametric statistics. Statistical interference. criteria To calculate and model probability distributions and stochastic variables. To estimate estimators and calculate test probabilities, and interpret the results. To test hypotheses using resampling and Q-tests. To use the associated symbolic language and mathematical formalism. To handle the mathematical concepts and ideas involved, plus their ranges and interrelations. Overall content The competency to apply mathematical thinking in relation to probability and stochastic and statistical phenomena, and to reflect on one s own learning in this connection. The competency to be able to follow, assess and carry out mathematical reasoning and proofs within probability theory and statistics. The competency to decode, interpret, differentiate between and link different mathematical representations within probability theory and statistics. The competency to understand, formulate, formalise and solve problems relating to stochastic phenomena. The competency to understand, evaluate, criticise and establish mathematical models of stochastic phenomena. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The competency to use digital tools in statistical surveys, as well as in model simulation and analysis. Probability theory: The classic mathematical formalisation and clarification of the concepts of probability, probability distribution, independence, conditional probability and probability distributions of finite, countable quantities and continuous distributions on the real axis. The most common distributions. Generating functions and branching processes. Teaching and working methods Statistics: Resampling techniques and non-parametric statistics. Introduction to likelihood-based statistical inference. Examples Lectures and exercises with brief student presentations and discussions of the material. Around 3-5 written assignments/tasks are included in the course, which may be prepared individually or in groups. 12

14 Form of examination Group examination or individual examination Moderation The course is assessed via an oral examination. The oral examination relates to written assignments/tasks undertaken during the course. The duration of the examination is 30 minutes, including assessment. Individual Internal Title Reflection project (Mathematics) Project 15 ECTS Of the academic standards of the project. Of the mathematical concepts involved, their range and relations. Of the mathematical symbolic language and formalism involved. criteria To analyse mathematical problems of advanced character using the forms of thinking, reasoning and representation that are characteristic of mathematics. To be able to place academic insights into perspective and illustrate the interaction of mathematics with historical, cultural and/or technological developments. To handle the relevant mathematical concepts. Overall content Teaching and working methods The competency to apply mathematical thinking to the subjects and issues of the project. The competency to be able to follow, assess and apply mathematical reasoning and proofs within the framework of the project. The competency to decode, interpret, differentiate between and link different mathematical representations in relation to the project. The competency to handle the symbols and mathematical formalism involved in the project. The competency to understand, formulate, formalise and solve problems within the framework of the project. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The project work is problem-oriented and exemplary. The problem statement of the project should involve a reflection on the nature and organisation of mathematics as a scientific discipline, professional subject and/or cultural discipline. The project s underlying mathematical material must include mathematical cases that shed light on and provide input towards an answer to the problem statement. Participant-controlled and problem-oriented group project work. The project work is assessed at an oral examination. The examination is based on the student s or group s project report. 13

15 Form of examination Group examination or individual examination Moderation The examination takes the form of a conversation between the students, the examiner and the moderator. Students must be examined on the basis of the whole project report. The examination must allow for the individual assessment of the students. Each student s grade is awarded on the basis of a combined individual assessment of both the project report and the oral examination. The duration of the examination is 30 minutes per student, including assessment. The project report must be provided with a cover page stating the title, project participant(s), supervisor(s), programme and semester. There must also be a table of contents, an abstract and a bibliography. The report must be between 20 and 50 standard pages in length, excluding bibliography and possible appendices. A standard page is 2,400 characters, including spaces. The project report must be written in an easily legible font. A report that fails to meet these requirements may be rejected as a basis for the project examination, and one examination attempt will be deemed to have been used up. Students whose project report has been rejected due to deviation from the size requirement will be given 24 hours to adapt the report to the required size, after which the project report is resubmitted. Group examination External Title Geometry (Mathematics) Course 10 ECTS General characteristics of regular curves and surfaces in R^3. Curvature concepts for regular surfaces in R^3. Abstract differential geometry. Examples of specific surfaces. criteria To be able to handle the concepts, ideas and entities of differential geometry. To apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the differential geometric structures studied. To apply mathematical analysis and linear algebra in differential geometry. The competency to apply mathematical thinking to geometrical structures and problems. The competency to be able to follow, assess and apply mathematical reasoning and proofs within geometry. The competency to decode, interpret, differentiate between and link different mathematical representations, particularly geometrical and algebraic representations. The competency to understand, formulate, formalise and solve problems within differential geometry. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The competency to use digital tools to investigate, solve and communicate issues in differential geometry. 14

16 Overall content Teaching and working methods Form of examination Regular curves in R^3. Regular surfaces in R^3. Curvature concepts for regular surfaces and their relations. Abstract surfaces and Riemannian metrics. Specific surfaces. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed via an oral examination. The examinee is given a take-home question three days before the oral examination. At the oral examination, the student first of all presents his or her answer to the oral question in a pre-prepared presentation of a maximum duration of 10 minutes. The student is then questioned about the curriculum generally for around 15 minutes. Group examination or individual examination Moderation The duration of the examination is 30 minutes, including assessment. Individual External Title Advanced mathematics (Mathematics) Optional course 5 ECTS One or more mathematical topics of an advanced character. criteria To handle the entities, symbols, terminology and results involved. To apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the subject. Problem-solving in relation to the subject. Overall content Teaching and working methods The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to be able to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The course addresses a selected area of mathematics, which is studied in detail. The main focus is on the conceptual foundation, the construction of a coherent theory and the detailed arguments for the theory s results. In each academic year, the selected area of the course is announced in advance by the Study Board. Lectures and exercises with brief student presentations and discussions of the material. Around 3-5 written assignments/tasks are included in the course, which may be prepared individually or in groups. The course is assessed via an oral examination. 15

17 Form of examination Group examination or individual examination Moderation The oral examination relates to written assignments/tasks undertaken during the course. The duration of the examination is 30 minutes, including assessment. Individual Internal Title Theme course A in Mathematics (Mathematics) Course Students who have passed a course in statistics, probability theory or geometry as part of their Bachelor study programmes will be given a syllabus in which this course can be included. 5 ECTS One or more selected mathematical topics. criteria To handle the entities, symbols, terminology and results involved. To be able to apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the subject. Problem-solving in relation to the subject. Overall content Teaching and working methods Form of examination Group examination or individual examination The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to read and understand mathematical texts within the subject, and to be able to communicate these both orally and in writing to various target groups. Dependent on the specific mathematical topic(s), and will be stated in the specific course description. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed through an oral presentation during the course and via approval of the answers to 3-5 assignment sets. Individual Title Theme course B in Mathematics (Mathematics) Course 16

18 Students who have passed a course in statistics, probability theory or geometry as part of their Bachelor study programmes will be given a syllabus in which this course can be included. 5 ECTS One or more selected mathematical topics. criteria To handle the entities, symbols, terminology and results involved. To be able to apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the subject. Problem-solving in relation to the subject. Overall content Teaching and working methods Form of examination Group examination or individual examination The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to read and understand mathematical texts within the subject, and to be able to communicate these both orally and in writing to various target groups. Dependent on the specific mathematical topic(s), and will be stated in the specific course description. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed through a 24-hour take-home written examination, followed by an oral examination of 30 minutes duration, including assessment. At the examination, the examinee first gives a presentation of approximately ten minutes duration on a task/subtask from the written assignment set. The examinee is then questioned on the curriculum generally for around 15 minutes. Individual Title Theme course C in Mathematics (Mathematics) Course Students who have passed a course in statistics, probability theory or geometry as part of their Bachelor study programmes will be given a syllabus in which this course can be included. 5 ECTS One or more selected mathematical topics. criteria To handle the entities, symbols, terminology and results involved. To be able to apply the associated symbolic language and mathematical 17

19 formalism. To read, understand and reproduce proofs in the context of the subject. Problem-solving in relation to the subject. Overall content Teaching and working methods The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to read and understand mathematical texts within the subject, and to be able to communicate these both orally and in writing to various target groups. Dependent on the specific mathematical topic(s), and will be stated in the specific course description. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed through a 24-hour take-home written examination, followed by an oral examination of 30 minutes duration, including assessment. Form of examination Group examination or individual examination At the examination, the examinee first gives a presentation of approximately ten minutes duration on a task/subtask from the written assignment set. The examinee is then questioned on the curriculum generally for around 15 minutes. Individual 5.3 Third semester The third semester consists of study activities in both Computer Science and Mathematics. Object The object of the third semester is to expand the student s knowledge of Computer Science and Mathematics. The student s specialisation is supported by course work in Computer Science. Study programme elements in the third semester The semester encompasses: Autumn start: Computer Science: Three optional courses in Computer Science (3*5 ECTS) Mathematics: Course in Geometry (10 ECTS) * Course in Advanced Mathematics (5 ECTS) Spring start: Computer Science: Three optional courses in Computer Science (3*5 ECTS) Mathematics: 18

20 Course in Fundamental Mathematical Structures (10 ECTS) Optional course in Probability Theory and Statistics (5 ECTS) * or Advanced Mathematics (5 ECTS) * In the case of content overlap between the qualifying Bachelor programme and the Mathematics portion of the Master s programme, students will be referred by the Head of Studies for Mathematics to either Theme Course A, B or C, or a modelling project. Title Language of instruction Project in Advanced Computer Science (Computer Science) English Project 15 ECTS In-depth knowledge of the theoretical, methodological and practical possibilities and problems associated with software development, using specific models from the core areas or optional topics. criteria To describe and reflect upon an independently-performed task that addresses a complex issue, using relevant solution models. To define and justify a selected solution model and independently plan and implement solutions, using relevant scientific literature at an advanced academic level. To master concepts, theories and methods on the basis of the literature, and be able to use these in a reflective manner to solve specific problems in computer science. Overall content Teaching and working methods Form of examination Group examination or individual examination Moderation To control computer science development situations that are complex and require new solution models. Project on a self-chosen issue relating to software development. The project must give the student an opportunity to describe and reflect upon an independently-performed task that addresses a complex issue. The project is normally carried out as a group project with 2-6 participants, but in special cases, the Head of Studies may grant permission for a student s project to be written or examined individually. The topic of the project work is chosen by the students in accordance with the requirements for learning outcomes and any defined topics and themes laid down by the Study Board. Oral project examination on the basis of the project report and any supplementary material (product or process documentation). Up to five minutes are allocated per student during the examination for a self-chosen, prepared presentation based on the joint project. The examination otherwise takes the form of a conversation between the student(s), the examiner and the moderator, under the direction of the examiner. During the examination, it must be ensured that all examinees are given an equal opportunity to demonstrate their knowledge of the joint project and their academic standard in relation to the expected learning outcomes. Group External 19

21 Title Geometry (Mathematics) Course 10 ECTS General characteristics of regular curves and surfaces in R^3. Curvature concepts for regular surfaces in R^3. Abstract differential geometry. Examples of specific surfaces. criteria To be able to handle the concepts, ideas and entities of differential geometry. To apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the differential geometric structures studied. To apply mathematical analysis and linear algebra in differential geometry. Overall content Teaching and working methods Form of examination Group examination or individual examination Moderation The competency to apply mathematical thinking to geometrical structures and problems. The competency to be able to follow, assess and apply mathematical reasoning and proofs within geometry. The competency to decode, interpret, differentiate between and link different mathematical representations, particularly geometrical and algebraic representations. The competency to understand, formulate, formalise and solve problems within differential geometry. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The competency to use digital tools to investigate, solve and communicate issues in differential geometry. Regular curves in R^3. Regular surfaces in R^3. Curvature concepts for regular surfaces and their relations. Abstract surfaces and Riemannian metrics. Specific surfaces. Lectures and exercises with brief student presentations and discussions of the material. The course is assessed via an oral examination. The examinee is given a take-home question three days before the oral examination. At the oral examination, the student first presents his or her answer to the oral question in a pre-prepared presentation of a maximum duration of 10 minutes. The student is then questioned about the curriculum generally for around 15 minutes. The duration of the examination is 30 minutes, including assessment. Individual External 20

22 Title Advanced mathematics (Mathematics) Optional course 5 ECTS One or more mathematical topics of an advanced character. criteria To handle the entities, symbols, terminology and results involved. To apply the associated symbolic language and mathematical formalism. To read, understand and reproduce proofs in the context of the subject. Problem-solving in relation to the subject. Overall content Teaching and working methods Form of examination Group examination or individual examination Moderation The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to be able to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The course addresses a selected area of mathematics, which is studied in detail. The main focus is on the conceptual foundation, the construction of a coherent theory and the detailed arguments for the theory s results. In each academic year, the selected area of the course is announced in advance by the Study Board. Lectures and exercises with brief student presentations and discussions of the material. Around 3-5 written assignments/tasks are included in the course, which may be prepared individually or in groups. The course is assessed via an oral examination. The oral examination relates to written assignments/tasks undertaken during the course. The duration of the examination is 30 minutes, including assessment Individual Internal Title criteria Fundamental Mathematical Structures (Mathematics) Compulsory course 10 ECTS Specific mathematical structures within set theory, topology and analysis, algebra and geometry. Common features of and differences between such structures. Different types of reasoning and proofs, and their importance. Construction and formalisation of such structures. 21

23 To recognise fundamental mathematical structures. To know and use symbols and other representations in accordance with the given formalism. To read, understand and reproduce proofs in the context of the structures studied. Overall content Teaching and working methods Form of examination Group examination or individual examination Moderation The competency to apply mathematical thinking in relation to the fundamental structures of the subject. The competency to be able to follow, assess and carry out mathematical reasoning and proofs. The competency to decode, interpret, differentiate between and link different mathematical representations. The competency to be able to decode and apply mathematical symbolic language within a given formalism, and to assess the strengths and weaknesses of an axiomatic system. The competency to be able to read and understand mathematical texts concerning the foundation of the subject and fundamental structures, and to communicate these both orally and in writing. Various fundamental, abstract mathematical structures and their interrelations. Introduction to formal logic, including the concept of a formal theory. Set theory, algebraic structures, metric and topological spaces, geometric structures and aspects of measure spaces Lectures and exercises with brief student presentations and discussions of the material. The course is assessed via an oral examination. The examinee is given a take-home question three days before the oral examination. At the oral examination, the student first presents his or her answer to the oral question in a pre-prepared presentation of a maximum duration of 10 minutes. The student is then questioned about the curriculum generally for around 15 minutes. The total duration of the examination is 30 minutes, including assessment. Individual External Title Probability Theory and Statistics (Mathematics) Optional course 5 ECTS General features of discrete and continuous probability distributions. The most common discrete and continuous probability distributions. Stochastic variables, independence, mean, variance, co-variance and correlation of stochastic variables. Statistical models, maximum likelihood principle, estimator tests, resampling and non-parametric statistics. Statistical interference. criteria To calculate and model probability distributions and stochastic variables. 22

24 To estimate estimators and calculate test probabilities, and interpret the results. To test hypotheses using resampling and Q-tests In using the associated symbolic language and mathematical formalism. To handle the mathematical concepts and ideas involved, plus their ranges and interrelations. Overall content The competency to apply mathematical thinking in relation to probability and stochastic and statistical phenomena, and to reflect on one s own learning in this connection. The competency to be able to follow, assess and carry out mathematical reasoning and proofs within probability theory and statistics. The competency to decode, interpret, differentiate between and link different mathematical representations within probability theory and statistics. The competency to understand, formulate, formalise and solve problems relating to stochastic phenomena. The competency to understand, evaluate, criticise and establish mathematical models of stochastic phenomena. The competency to be able to read and understand mathematical texts within the subject, and to communicate these both orally and in writing. The competency to use digital tools in statistical surveys, as well as in model simulation and analysis. Probability theory: The classical mathematical formalisation and clarification of the concepts of probability, probability distribution, independence, conditional probability and probability distributions of finite, countable quantities and continuous distributions on the real axis. The most common distributions. Generating functions and branching processes. Teaching and working methods Form of examination Group examination or individual examination Moderation Statistics: Resampling techniques and non-parametric statistics. Introduction to likelihood-based statistical inference. Examples Lectures and exercises with brief student presentations and discussions of the material. Around 3-5 written assignments/tasks are included in the course, which may be prepared individually or in groups. The course is assessed via an oral examination. The oral examination relates to written assignments/tasks undertaken during the course. The duration of the examination is 30 minutes, including assessment. Individual Internal 23

25 5.4 Fourth semester thesis The thesis corresponds to 30. The thesis is written on the basis of Computer Science, and may be interdisciplinary. The thesis is written in the teaching and examination language of Computer Science. The Head of Studies must approve the thesis statement, and will at the same time set a final date for the submission for the thesis, pursuant to the regulations for examination registration. The thesis must be accompanied by a summary in a foreign language. If the thesis is written in a foreign language other than Norwegian or Swedish, the summary may be written in Danish. The thesis work, which usually takes place in groups, concludes the study programme; the Study Board may however grant an exemption from this requirement where this is justified by exceptional circumstances, or if one or more previous study programme elements have not been passed. Research-based knowledge in selected academic areas, and an understanding of and reflection on the position of the student s own thesis study in the academic field. Identification of scientific issues and a critical approach to scientific knowledge. of the academic genre and the academic target group, and an understanding of the overall communication situation of which the thesis is a part. in mastering and applying scientific theories and methods in work on a defined, academic and relevant issue. in analysing, categorising, discussing, reasoning, reflecting and evaluating on a scientific basis. The ability to critically evaluate and select sources, literature, theory and methods. The ability to discuss and participate in academic debate with representatives of the academic discourse community. The ability to write in accordance with academic norms and for an academic target group. The competency to independently initiate, manage and complete a long-term process of academic study and writing. The competency to take responsibility for one s own academic and linguistic development and specialisation. Size: Between 20 and 100 standard pages of 2,400 characters each (incl. spaces), excluding the front page, table of contents, bibliography and any appendices. Oral examination In an oral examination of the thesis, the duration of the examination, including assessment, is set as follows: 1 examinee = 40 minutes 2 examinees = 70 minutes 3 examinees = 85 minutes 4 examinees = 100 minutes 5 examinees = 115 minutes 6 examinees = 130 minutes In the assessment of the thesis, the following forms of assessment may be used: 1. on the basis of the thesis alone, or 2. Combined assessment of the thesis and an oral examination. The student selects the examination form. The rector may however approve that regulations are laid down in the curriculum stipulating the use of a fixed form of assessment. The form of assessment may be made dependent on whether the thesis is written by a group or by an individual. If the thesis is written by a group, it must be individualised on submission if one or more of the students in the group wish to have the thesis assessed on the basis of the thesis alone (form 1). The individualisation must be genuine and show which students have been mainly responsible for the individual sections. If the thesis is not individualised, an oral thesis examination 24

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